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INPE National Institute for Space Research
So Jos dos Campos SP Brazil July 26-30, 2010
ON THE EFFECT OF A PARALLEL RESISTOR IN THE CHUAS CIRCUIT
Flavio Prebianca1, Gustavo Lambert
2, Holokx A. Albuquerque
3, Rero M. Rubinger
4
1 Santa Catarina State University, Joinville, Brazil, [email protected]
2 Santa Catarina State University, Joinville, Brazil, [email protected]
3 Santa Catarina State University, Joinville, Brazil, [email protected]
4 Federal University of Itajub, Itajub, Brazil, [email protected]
keywords: Chaotic Dynamics, Applications of NonlinearSciences, Time series analysis, Applications in
Engineering and Nanoscience.
We studied the dynamical behavior of a Chuas
circuit with parallel resistor. Our studies are based on a
recent work of Braga, Mello, and Messias [1], where theauthors analytically studied the governing nonlinear
equations, where the nonlinearity of the diode is a cubic
function, and the intrinsic inductor resistance was
omitted. In our case, we numerically studied the
complete set of equations, based on the Chuas circuit
model [2], with a piecewise-linear function for the
nonlinearity, and adding a resistor in parallel with the
inductor. Our aim is to obtain the global bifurcation
behaviors constructing two-dimensional parameter spaces
of the model with the largest Lyapunov exponent method.
We also realized the experimental circuit, and we obtained
the experimental phase portraits (attractors) for three
parameters of the system.
The set of equations that describes the Chuas circuit
with parallel resistor is given by
( ) ( ) ( )11112
1
1 CviRCvvdt
dvv d==&
,
( ) ( ) 22211
2 CiRCvvdtdvv L+== & , (1)
( )LriLvdt
dii LL
L
L == 2& ,
with the dynamical variablesLi , the current across the
inductor,1v and 2v , the voltages across the capacitors 1C
and2C , respectively. The nonlinearity is given by
( ) ( )( )bvbvmmvmvid ++= 110121101 , and
( ) PP RRR+= , where PR is the parallel resistor whichmodifies the standard Chuas circuit, i.e., for 1= [2].
The numerical study carried out in this work consists
of to calculate the largest Lyapunov exponent,
numerically solving the Eqs. (1) with fourth-order Runge-
Kutta method with time step equal to 110 , for each pair of
parameters ( )LrR, . The range of parameter values wasdiscretized in a mesh of 500500 points equally spaced.We identify for each largest Lyapunov exponent a color,
varying continuously from black (zero exponent), passing
through yellow (positive exponent), up to red (positive
exponent).
Figure 1 shows the two-dimensional parameter space
for the parameters ( )LrR, in Eqs. (1). Black regionsrepresent periodic behaviors, and the yellowish regions
represent chaotic behaviors. The blue color represents the
divergence of Eqs. (1). Inside the chaotic regions, we can
observe the existence of immersed periodic structures,
represented by the black regions inside of the yellowishregions. Figure 2 shows the attractors for parameter values
in the two green marks of Fig. 1. We observe attractors
with periodic (black regions) and chaotic behaviors
(yellow regions).
Figure 1 Global view of the ( )LrR, parameter space of Eqs. (1). The
axes are in resistance units (Ohms). Black color indicates periodic
behavior, yellow one indicates chaotic behavior. The white regionindicates fixed points. The blue region indicates divergence of Eqs.
(1). The black region marked with FP, is another fixed points region.
The points a and b locate the parameter values of the attractors
shown in Fig. 2.
Periodic structures embedded in chaotic regions inthe Chuas circuit were reported in recent theoretical and
FP
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(a) (b)
(c) (d)
experimental works [2-5], where the Chuas circuit is
modeled by Eqs. (1), for 1= , and with cubic
nonlinearity [4] or piecewise-linear function [2,3,5]. In
Ref. [5], we observed periodic structures organize
themselves in a single spiral structure that coils up around
a chaotic focal point in ( )LrR, parameter space, like thatreported in Ref. [2]. However, in the Chuas circuit with
parallel resistor here studied, i.e., Eqs. (1), the spiralstructure was destroyed, and a large fixed points region
emerged, shown in Fig. 1 as a large black region marked
with FP.
In Fig. 3, we show four experimental attractors of the
implemented Chuas circuit with parallel resistor. The
experimental parameters were: (a) 0.1362=R , 2.1=Lr
, 0.2217=PR ; (b) 0.1375=R , 0.1=Lr ,
0.2261=PR ; (c) 0.1384=R , 0.8=Lr ,
0.2218=PR ; (d) 0.1434=R , 5.1=Lr ,
0.2238=PR .
Figure 2 Theoretical (a) periodic and (b) chaotic attractors for the
parameter values of the two green marks a and b in Fig. 1. The
axes are in voltage units (Volts).
Figure 3 Four experimental attractors for the implemented chaotic
circuit. See the similarities between the attractors (a) and (b) withthe theoretical attractors (a) and (b) in Fig. 2.
In Figs. 3(a) and 3(b), we can observe the similarities
between the experimental attractors with the attractors
obtained by numerical solutions of Eqs. (1) shown in
Figs. 2(a) and 2(b). Two other experimental attractors,
Figs. 3(c) and 3(d), were found in the implemented
circuit. Those attractors present an interesting feature,
periodic left side connected with chaotic right side.
A two-dimensional parameter space, using the
largest Lyapunov exponent codified in a continuous range
of colors, for the Chuas circuit with parallel resistor was
reported. With that modification, we observed the
disappearance of the spiral structure and the appearance of
a fixed points region. In the experimental implementation
of the circuit, we observed a good agreement between
theoretical and experimental attractors. Two new
experimental chaotic attractors were observed, with one
side periodic and other one chaotic.
ACKNOWLEDGMENTS
The authors thank Conselho Nacional de
Desenvolvimento Cientfico e Tecnolgico-CNPq, and
Fundao de Apoio Pesquisa Cientfica e Tecnolgica do
Estado de Santa Catarina-FAPESC, Brazilian agencies, for
the financial support. FP and HAA also thank Prof.
Ricardo A. de Simone Zanon for his support in themeasuring apparatus.
References
[1] D.C. Braga, L.F. Mello, and M. Messias, Bifurcation
Analysis of a Van der Pol-Dffing Circuit with Parallel
Resistor, Math. Probl. Engin. Vol. 2009, 149563,
2009.
[2] H.A. Albuquerque, R.M. Rubinger, and P.C. Rech,
Self-similar Structures in a 2D Parameter-pace of an
Inductorless Chuas Circuit, Phys. Lett. A Vol. 372,
pp. 4793-4798, 2008.
[3] E.R. Viana Jr., R.M. Rubinger, H.A. Albuquerque,
A.G. de Oliveira, and G.M. Ribeiro, High-resolutionParameter Space of an Experimental Chaotic Circuit,
to appear, Chaos, June 2010.
[4] C. Stegemann, H.A. Albuquerque, and P.C. Rech,
Some Two-dimensional Parameter Spaces of a Chua
System with Cubic Nonlinearity, to appear, Chaos,
June 2010.
[5] H.A. Albuquerque, and P.C. Rech, Spiral Periodic
Structures inside Chaotic Region in Parameter-Space
of a Chua Circuit, not published, 2009.